From Periodicity to Non-Repetitive Order: A Physico-Mathematical Analysis of tessellations up to Penrose Pattems from the Ontosemiotic Approach

Abstract

This article analyzes the conceptual, geometric, and physico-mathematical evolution of planar tessellations, from classical periodic patterns to the complex aperiodic Penrose tilings. The main objective is to understand how the transition from periodic to quasiperiodic order can be addressed didactically through the Ontosemiotic Approach to mathematical knowledge and instruction (EOS), integrating epistemic, semiotic, cognitive, and institutional dimensions. The methodology employed corresponds to a qualitative, interpretative study focused on a didactic experience conducted with students from the Bachelor's Degree in the Teaching of Mathematics and Physics. Progressive geometric tasks were designed and analyzed, focusing on periodic, aperiodic, and Penrose tessellations, using reflective didactic configurations guided by EOS principles. Among the results, it is noteworthy that students were able to articulate geometric, symbolic, and verbal representations, identify properties of quasiperiodicity and non-conventional symmetry, and reconceptualize the notion of order in non-periodic contexts. Likewise, semiotic conflicts and conceptual reorganizations were identified, particularly in the transition from conventional tessellations to structures like those of Penrose. Regarding the conclusions, it is argued that the study of tessellations—especially aperiodic ones—not only enhances mathematical understanding of complex structures, but also fosters interdisciplinary educational processes that connect geometry, solid-state physics, and advanced didactics. This approach offers an effective path to prepare future educators capable of interpreting, modeling, and communicating non-standard geometric phenomena within the teaching of modern mathematics.